A151347 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1)}.

1, 0, 1, 1, 3, 8, 19, 65, 177, 611, 1928, 6648, 22928, 80851, 292343, 1063611, 3957406, 14818681, 56339994, 215943994, 836246604, 3265240671, 12848804154, 50936668789, 203235590343, 816070826188, 3295317218038, 13379003847708, 54588942258042, 223782828113783, 921414594957514, 3809576782931810

Offset: 0

Author

Manuel Kauers, Nov 18 2008

Keywords

• nonn
• walk

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..500
• A. Bostan, K. Raschel, B. Salvy, Non-D-finite excursions in the quarter plane, J. Comb. Theory A 121 (2014) 45-63, Table 1 Tag 35, Tag 39.
• M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.
• Colin Defant, Motzkin Intervals and Valid Hook Configurations, arXiv:1904.10451 [math.CO], 2019.
• Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019.

Programs

Mathematica

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]